Part I: Lesson

In the prior lesson, we introduced the rules of inference for conditional statements known as Modus Ponens and Modus Tollens. In today’s lesson, we’ll be looking back at these rules of inference to see what happens when the wrong logical steps are taken during the use of these rules.

There are two major mistakes that can occur in the use of the rules listed above; each of these mistakes corresponds to one of the rules. It is also worth mentioning that both mistakes involve the production of an incorrect second premise of each rule of inference.

i. Affirming the Consequent

The first of these two mistakes is called Affirming the Consequent; this name should be revealing once it is noted that Modus Ponens is alternatively referred to as Affirming the Antecedent. Modus Ponens takes the form:

1. If p, then q
2. p
3. therefore q

or (for only categorical propositions)

1. All A’s are B’s.
2. C is an A / All C’s are A’s.
3. Therefore, C is a B / All C’s are B’s.

This new fallacy, Affirming the Consequent, occurs when an argument takes the following form:

1. If p, then q 
2. q
3. therefore p 

or (for only categorical propositions) 

1. All A’s are B’s.
2. C is a B / All C’s are B’s.
3. Therefore, C is an A / All C’s are A’s.

Notice that for the cases with hypothetical propositions, the fallacy switches the second premise and the conclusion of Modus Ponens. This is a mistake, as the conditional only asserts that the truth of p implies that of q; it does not assert the opposite, that q implies p, which the mistaken form needs to be a valid argument. 
For the cases with only categorical propositions, the mistake can easily be seen when we realize that there might be some B’s that are not A’s. Therefore, just because something is a B, that does not mean that that thing is an A. This can easily be seen with Euler Circles:

This set of Euler circles says both that (1) all A’s are B’s and that (2) C is a B. However, it shows that (1) and (2) could both be the case while C is not an A. This is contradictory with the conclusion, (3). Thus, Affirming the Consequent for only categorical propositions is also fallacious.

ii. Denying the Antecedent

The second mistake, Denying the Antecedent, relates to Modus Tollens, also known as Denying the Consequent, in a similar way to how Asserting the Consequent relates to Modus Ponens. Modus Tollens has the form:

1. If p, then q 
2. not q
3. therefore not p.

or (for only categorical propositions) 

1. All A’s are B’s.
2. C is not a B / No C’s are B’s.
3. Therefore, C is not an A / No C’s are A’s.

An argument, making the mistake of Denying the Antecedent, has the form:

1. If p, then q
2. not p
3. therefore, not q 

or (for only categorical propositions) 

1. All A’s are B’s.
2. C is not an A / No C’s are A’s.
3. Therefore, C is not a B / No C’s are B’s.

The reason that this is a mistake is that the conditional links the truth of “p” to that of “q”, such that all instances in which “p” is true also have “q” as being true. However, there certainly can be cases in which q is the case without p being the case (examples will make this clearer). It is important to understand these mistakes: an argument which commits one of these mistakes is invalid.For the cases with only categorical propositions, the mistake can easily be seen when we again realize that there might be some B’s that are not A’s. Therefore, just because something is not an A does not mean that that thing is not a B. This can easily be seen with the same Euler Circles:

This set of Euler’s circles says both that (1) All A’s are B’s, and that (2) C is not an A. However, this is a case where both (1) and (2) are the case, while (3) is not the case. That is, C is a B! Therefore, Denying the Antecedent is fallacious. 

In using Affirming the Consequent and Denying the Antecedent true premises no longer lead to assuredly true conclusions. Instead, the conclusions reached are either true, but only by happenstance, or they are in fact false. Mistakes in reasoning of this kind are known as fallacies. Fallacies can take two forms: there are formal fallacies, like the ones shown in this lesson, which have to do with the formal structure of arguments and make these arguments invalid; and there are informal fallacies, in which problems in relation to the propositional content of the premises lead to false conclusions. Informal fallacies will be dealt with in later lessons.